p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8:2SD16, C42.239C23, C4:C4.61D4, C8:4Q8:2C2, C8:2C8:15C2, (C2xC8).91D4, (C2xQ8).56D4, C4:SD16:35C2, C4.D8:17C2, C8:4D4.13C2, C2.7(C8:2D4), C4.82(C2xSD16), C4:C8.184C22, C4.70(C8:C22), (C4xC8).142C22, (C4xQ8).44C22, C2.10(C4:SD16), C4:1D4.35C22, C2.13(D4.4D4), C22.200(C4:D4), (C2xC4).24(C4oD4), (C2xC4).1274(C2xD4), SmallGroup(128,420)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8:2SD16
G = < a,b,c | a8=b8=c2=1, bab-1=a3, cac=a-1, cbc=b3 >
Subgroups: 272 in 92 conjugacy classes, 34 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, Q8, C23, C42, C42, C4:C4, C4:C4, C2xC8, C2xC8, D8, SD16, C2xD4, C2xQ8, C4xC8, C8:C4, D4:C4, C4:C8, C4:C8, C4:C8, C4xQ8, C4:1D4, C2xD8, C2xSD16, C4.D8, C8:2C8, C8:4Q8, C4:SD16, C8:4D4, C8:2SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2xD4, C4oD4, C4:D4, C2xSD16, C8:C22, C4:SD16, C8:2D4, D4.4D4, C8:2SD16
Character table of C8:2SD16
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | |
size | 1 | 1 | 1 | 1 | 16 | 16 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
ρ6 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | linear of order 2 |
ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 2 |
ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | linear of order 2 |
ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | complex lifted from C4oD4 |
ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | complex lifted from C4oD4 |
ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | √-2 | 0 | -√-2 | √-2 | 0 | -√-2 | complex lifted from SD16 |
ρ16 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -√-2 | 0 | √-2 | -√-2 | 0 | √-2 | complex lifted from SD16 |
ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | √-2 | 0 | √-2 | -√-2 | 0 | -√-2 | complex lifted from SD16 |
ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -√-2 | 0 | -√-2 | √-2 | 0 | √-2 | complex lifted from SD16 |
ρ19 | 4 | -4 | -4 | 4 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8:C22 |
ρ20 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8:C22 |
ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8:C22 |
ρ22 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | 0 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 0 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 25 39 63 24 45 53 10)(2 28 40 58 17 48 54 13)(3 31 33 61 18 43 55 16)(4 26 34 64 19 46 56 11)(5 29 35 59 20 41 49 14)(6 32 36 62 21 44 50 9)(7 27 37 57 22 47 51 12)(8 30 38 60 23 42 52 15)
(2 8)(3 7)(4 6)(9 46)(10 45)(11 44)(12 43)(13 42)(14 41)(15 48)(16 47)(17 23)(18 22)(19 21)(25 63)(26 62)(27 61)(28 60)(29 59)(30 58)(31 57)(32 64)(33 51)(34 50)(35 49)(36 56)(37 55)(38 54)(39 53)(40 52)
G:=sub<Sym(64)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,25,39,63,24,45,53,10)(2,28,40,58,17,48,54,13)(3,31,33,61,18,43,55,16)(4,26,34,64,19,46,56,11)(5,29,35,59,20,41,49,14)(6,32,36,62,21,44,50,9)(7,27,37,57,22,47,51,12)(8,30,38,60,23,42,52,15), (2,8)(3,7)(4,6)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,48)(16,47)(17,23)(18,22)(19,21)(25,63)(26,62)(27,61)(28,60)(29,59)(30,58)(31,57)(32,64)(33,51)(34,50)(35,49)(36,56)(37,55)(38,54)(39,53)(40,52)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,25,39,63,24,45,53,10)(2,28,40,58,17,48,54,13)(3,31,33,61,18,43,55,16)(4,26,34,64,19,46,56,11)(5,29,35,59,20,41,49,14)(6,32,36,62,21,44,50,9)(7,27,37,57,22,47,51,12)(8,30,38,60,23,42,52,15), (2,8)(3,7)(4,6)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,48)(16,47)(17,23)(18,22)(19,21)(25,63)(26,62)(27,61)(28,60)(29,59)(30,58)(31,57)(32,64)(33,51)(34,50)(35,49)(36,56)(37,55)(38,54)(39,53)(40,52) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,25,39,63,24,45,53,10),(2,28,40,58,17,48,54,13),(3,31,33,61,18,43,55,16),(4,26,34,64,19,46,56,11),(5,29,35,59,20,41,49,14),(6,32,36,62,21,44,50,9),(7,27,37,57,22,47,51,12),(8,30,38,60,23,42,52,15)], [(2,8),(3,7),(4,6),(9,46),(10,45),(11,44),(12,43),(13,42),(14,41),(15,48),(16,47),(17,23),(18,22),(19,21),(25,63),(26,62),(27,61),(28,60),(29,59),(30,58),(31,57),(32,64),(33,51),(34,50),(35,49),(36,56),(37,55),(38,54),(39,53),(40,52)]])
Matrix representation of C8:2SD16 ►in GL8(F17)
16 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 16 | 2 | 0 | 0 | 0 | 0 |
2 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
1 | 16 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 5 | 5 | 7 | 7 |
0 | 0 | 0 | 0 | 12 | 12 | 0 | 10 |
0 | 0 | 0 | 0 | 5 | 12 | 0 | 0 |
0 | 0 | 0 | 0 | 12 | 0 | 12 | 0 |
5 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
12 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
7 | 7 | 7 | 10 | 0 | 0 | 0 | 0 |
7 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 16 | 16 | 1 | 2 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 0 | 1 | 1 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
15 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
16 | 16 | 16 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 16 | 16 |
G:=sub<GL(8,GF(17))| [16,0,2,1,0,0,0,0,0,16,0,16,0,0,0,0,16,16,1,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,5,12,5,12,0,0,0,0,5,12,12,0,0,0,0,0,7,0,0,12,0,0,0,0,7,10,0,0],[5,12,7,7,0,0,0,0,5,5,7,0,0,0,0,0,0,0,7,12,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,16,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,1,0,0,0,0,0,2,0,1],[1,0,15,16,0,0,0,0,0,16,0,16,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,16] >;
C8:2SD16 in GAP, Magma, Sage, TeX
C_8\rtimes_2{\rm SD}_{16}
% in TeX
G:=Group("C8:2SD16");
// GroupNames label
G:=SmallGroup(128,420);
// by ID
G=gap.SmallGroup(128,420);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,288,422,387,352,1123,136,2804,718,172]);
// Polycyclic
G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^3,c*a*c=a^-1,c*b*c=b^3>;
// generators/relations
Export